The principle of the determination of dimensional characteristics of particle distribution from measurements of forward light scattering has been recounted many times. Briefly, the particle distribution is illuminated by a coherent light beam, its far-field diffraction pattern is formed, and the light in this pattern is spatially modulated before detection in such a manner that the output signal is proportional to some desired characteristic of the distribution. As was taught by Stetter in STAUB, 30, 225, 1952 in the article entitled "Uber integrale optische Staubmessung" a spatial modulation of the far-field diffraction pattern in accordance with the reciprocal of the radius from the center of the pattern, as by using a split shaped aperture in a spatial filter, gives a total flux transmission proportional to the volume of the particles. Stetter also suggests that the difficulty of filtering out the light beam from the central area of the pattern can be compensated for to some degree by expanding the slit at its centrally located end.
In U.S. Pat. No. 3,809,478 issued to John Henry Talbot on May 7, 1974, there is set forth an equation for the transfer function of a filter for determining the n'th moment of a particel distribution by number. That equation indicates that the transmission factor of the filter at any point should be directly related to the (2-n)th power of the radial dimension of that point. That transfer function, of course, gives the same shape for a filter aperture for determining the total volume of the particles as suggested by Stetter in his 1952 publication and the Talbot description suggests that the equation adequately defines filters for determining other moments.
As set forth in U.S. Pat. No. 3,873,206 issued to William Leslie Wilcock on Mar. 25, 1975, it is necessary to not only compensate for the inability to utilize the central portion of the far-field diffraction pattern because of the presence of the incident light beam in that area, but it is also necessary to compensate for the outer limits of the filter which is used to spatially modulate the diffraction pattern. Compensation relating to the outer limits of the filter is necessary because of vignetting by the collector lens or loss of light flux which occurs because of the finite size of the filter mask as dictated by the size of the collecting lens receiving the scattered light.
The method of determining the dimensional characteristics of particle distribution from measurements of forward scattered light is most simply understood in the context of particles of spherical shape. For such a particle whose radius "a" is sufficiently large compared with the wavelength .lambda. of the incident beam, the radial intensity distribution in the far-field diffraction pattern is EQU I(w,a) = Ek.sup.2 a.sup.4 [J.sub.1 (kaw)/kaw].sup.2 ( 1)
where E is the flux per unit area in the incident beam, k = 2.pi./.lambda., w = sin .theta. with .theta. the angle relative to the direction of the incident beam, and J.sub.1 is the first-order Bessel function of the first kind. If this distribution is spatially modulated by a function T(w), the integrated signal from a particle of radius "a" is EQU S(a) = C.sub.1 .intg..sub.o.sup.1 I (w,a) T(w) 2.pi.w dw EQU = C.sub.2 a.sup. 2 .intg..sub.o .sup.1 w.sup.- T (w) J.sub.1.sup.2 (kaw) dw (2)
where the C's are instrumental constants.
Suppose now that it is desired to have the signal S proportional to the nth power of the radius of the particle, i.e., EQU S .infin. a.sup.n. (3)
It is clear from (2) that (3) is secured if EQU .intg..sub.o.sup.1 w.sup.-.sup.1 T(w) J.sub.1.sup.2 (kaw)dw = K.sub.n a.sup.n.sup.-2 ( 4)
where K.sub.n is a constant involving k. This integral equation (4) defines the required modulation function T(w), and the essential problem of the method is to find a solution of (4) which is valid for the range of "a" which is of interest, but which also satisfies the physical constraints of a practical instrument.
To illustrate this problem consider the modulating function EQU T.sub.3 (w) = b.sub.3 w.sup.- 1, 0 .ltoreq. w .ltoreq. 1 (5)
where b.sub.3 is a constant. Substitution in the left hand side of (4) gives EQU .intg..sub.o.sup.1 w.sup.-.sup.1 T.sub.3 (w) J.sub.1.sup.2 (kaw) dw = b.sub.3 .intg..sub.o.sup.1 w.sup.-.sup.2 J.sub.1.sup.2 (kaw) dw = b.sub.3 ka .intg..sub.o.sup.ka x.sup.-.sup.2 J.sub.1.sup.2 (x) dx
where x has been used to replace kaw. For values of particle diameter "a" which are large enough for the expression (1) to be valid, .intg..sub.o.sup.ka x.sup.- 2 J.sub.1.sup.2 (x) dx is negligibly different from .intg..sub.o.sup..infin. x.sup.-.sup.2 J.sub.1.sup.2 (x) dx = 4/3.pi.. Hence EQU .intg..sub.o.sup.1 w.sup.-.sup.1 T.sub.3 (w) J.sub.1.sup.2 ( kaw) dw = [(4b.sub.3 k.pi.)/3] a,
and comparing this with (4) it appears that the modulating function of (5) is a solution of (4) for n = 3, i.e. that it leads to a response proportional to the third power of the radius, and so proportional to the volume, of the particle. In a similar way it can be shown that the family of modulating functions EQU T.sub.n (w) = b.sub.n w.sup.2.sup.-n, 0 .ltoreq. w .ltoreq. 1 (6)
where n = 1, 2, . . . lead to responses proportional to the first, second, . . . powers of the radius of the particle.
Unfortunately these modulating functions do not represent possible solutions for a practical instrument, because they require T(w) to have non-zero values over the whole range of w from zero to unity. Now the central region of the diffraction field, where w is near zero, is not usable because it contains the light from the unscattered beam, which must as far as possible be excluded from the measurement. In consequence T(w) must be zero for values of w less than some value w.sub.i which is determined by the optical characteristics of the illuminating beam and the acceptable level of unscattered light. Similarly, the geometrical configuration of the instrument sets an upper limit w.sub.o to the angular field over which the diffracted light can be collected, which means that T(w) must be zero for values of w greater than w.sub.o. In practice the choice of modulating functions is thus restricted to those which satisfy the condition that T(w) is non-zero only for w.sub.i .ltoreq. w.ltoreq. w.sub.o, so that in place of (4) the integral equation defining T can be written ##EQU1##
Experience shows that the family of functions represented by (6) is never an optimum solution of (7), and for some values of n it does not even provide useful approximation in the sense that there is no reasonable range of particle sizes for which S is approximately proportional to a.sup.n. The most favorable case is with n = 2, which corresponds to T = constant. The further n is from 2 the further the response departs from the desired proportionality to a.sup.n. The droop with increasing "a" is due to the inner limit w.sub.i, because a larger proportion of the light scattered by larger particles is in the unusable central part of the field. Similarly the droop for smaller values of "a" is due to the outer limit w.sub.o because a larger proportion of light from smaller particles is scattered outside the angular limits of the instrument.
It is possible to obtain greatly improved solutions of (7) by dividing the range from w.sub.i to w.sub.o into two or more parts in each of which T takes the form of a polynomial in w (or .theta.), viz. EQU T (w) = .alpha..sub.1 + .beta..sub.1 w + .gamma..sub.1 w.sup.2 +. . . , w.sub.i &lt;w 21.sub.1, EQU = .alpha..sub.2 + .beta..sub.2 w + .gamma..sub.2 w .sup.2 + . . . , w.sub.1 &lt;w &lt; w.sub.2, EQU = .alpha..sub.p + .beta..sub.p w + .gamma..sub.p w.sup.2 + . . . , w.sub.p-1 &lt;w&lt; w.sub. o,
where .alpha..sub.1, .alpha..sub.2 . . . , .beta..sub.1, .beta..sub.2 . . . , .gamma..sub.1, .gamma..sub.2 . . . are constants. Optimum values of these coefficients can be chosen, for example, by requiring that the root-mean-square departure from the desired power of particle radius "a" be minimized over a selected range of particle radius "a".
The solution of this problem depends on the recognition of a new concept regarding the modulating function "T". Hitherto the modulation of the diffraction pattern has been thought of in terms of optical spatial filtration (or its electrical equivalent); the intensity is altered at particular points by passing the diffracted flux through a filter whose transmission is characterized by the chosen function "T". Since the transmission is essentially zero or positive this process confines the choice of "T" to functions which are everywhere positive. If "T" can take values which are negative a greatly enlarged choice of functions becomes available for the solution of (7). This is the essence of the advance which is exemplified by this invention and which has allowed the development of an improved method and means for determining dimensional characteristics of a particle distribution for the cases of n&gt;2 over a broad range of particle sizes and for sharpening the cut-off of the response characteristic at the large particle end of the characteristic for broad and limited particle size ranges.
Thus, it has been found that the teaching of others provides an adequate method and means for obtaining some of the lower moments, such as the second and third moments, of a particle distribution by number, which would be respectively proportional to the summations of the second and third powers of the diameter of the particles in the collection if the range of particle sizes is limited. However, even the third moment cannot be satisfactorily obtained for a very broad range of particle sizes with a single aperture filter as taught by others, because the effect of the inner limit of the mask is increasingly severe for the higher moments. It is therefore an object of this invention to provide a method and means for improving the analysis of a collection of particles to determine particular dimensional characteristics of the particles in the collection. More particularly, it is an object of this invention to provide a method and means for improving the determination of the third and fourth moments by providing means for compensating for the physical limits of the filter and extending the response in the large particle region as well as sharpening the cut-off of the response characteristic in that region.